Theorem gagrp 19141

Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
gagrp	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Proof of Theorem gagrp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2733	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2733	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 19140	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simplbi 499	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
6	5	simpld 496	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   × cxp 5670  ⟶wf 6531  ‘cfv 6535  (class class class)co 7396  Basecbs 17131  +_gcplusg 17184  0_gc0g 17372  Grpcgrp 18806   GrpAct cga 19138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-ga 19139

This theorem is referenced by:  gafo  19145  gass  19150  galcan  19153  gacan  19154  gapm  19155  gaorber  19157  gastacl  19158  galactghm  19256  sylow2alem2  19470